張る is to stretch. You're stubborn and want to stretch out what you're doing right now. Although you're stubborn, you keep doing it. You persevere. That's why this word is to persevere or to do one's best.

This word gets its reading from the kanji 頑 and the verb 張る put together, but the 張る part gets rendaku'd to ばる. Here's a mnemonic to help you remember that rendaku:

When you have to persevere, you always scream baa (ば)! Not sure why, you just think when someone has to do one's best, screaming "baa!" is a great motivator. Kinda like how weightlifters and tennis players grunt when they're doing their best, you know? Baa!